At a picnic there were 3 times as many adults as children and twice as many women as men. If...

1. Translate the problem requirements: We need to clarify that "3 times as many adults as children" means \(\mathrm{adults = 3 \times children}\), "twice as many women as men" means \(\mathrm{women = 2 \times men}\), and we want to find the number of men in terms of the total number x
2. Set up relationships using simple variables: Choose men as our base variable since that's what we're solving for, then express all other groups in terms of men
3. Express the total in terms of our base variable: Add up all groups (men + women + children) to get an expression equal to x
4. Solve for men in terms of x: Rearrange the equation to isolate the number of men and express it as a fraction of x

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what each phrase means in everyday language:

"3 times as many adults as children" means: If there are some number of children, then the number of adults is 3 times that amount. For example, if there are 5 children, there would be 15 adults.

"Twice as many women as men" means: If there are some number of men, then the number of women is 2 times that amount. For example, if there are 10 men, there would be 20 women.

We also know that adults consist of both men and women (no other categories), and the total of everyone (men + women + children) equals x.

Our goal is to find how many men there were, expressed as some fraction of x.

Process Skill: TRANSLATE - Converting the English relationships into mathematical understanding

2. Set up relationships using simple variables

Since we want to find the number of men, let's use that as our starting point. We'll call the number of men "M".

From our translation:

• Number of men = M
• Number of women = 2M (twice as many women as men)
• Number of adults = \(\mathrm{M + 2M = 3M}\) (men plus women)

Now, since there are "3 times as many adults as children":

• If adults = 3M, and \(\mathrm{adults = 3 \times children}\)
• Then: \(\mathrm{3M = 3 \times children}\)
• So: \(\mathrm{children = M}\)

This gives us:

• Men: M
• Women: 2M
• Children: M
• Adults: 3M (which equals Men + Women = \(\mathrm{M + 2M}\) ✓)

3. Express the total in terms of our base variable

The total number of people at the picnic is everyone added together:

\(\mathrm{Total = Men + Women + Children}\)

Substituting our expressions:

\(\mathrm{Total = M + 2M + M = 4M}\)

Since we're told the total equals x:

\(\mathrm{4M = x}\)

4. Solve for men in terms of x

We have the equation: \(\mathrm{4M = x}\)

To find M (the number of men) in terms of x, we divide both sides by 4:

\(\mathrm{M = x/4}\)

Let's verify this makes sense:

• Men: \(\mathrm{x/4}\)
• Women: \(\mathrm{2(x/4) = x/2}\)
• Children: \(\mathrm{x/4}\)
• Total: \(\mathrm{x/4 + x/2 + x/4 = x/4 + 2x/4 + x/4 = 4x/4 = x}\) ✓

Also checking our original relationships:

• Adults = \(\mathrm{x/4 + x/2 = 3x/4}\)
• Children = \(\mathrm{x/4}\)
• Is \(\mathrm{3x/4 = 3 \times (x/4)}\)? Yes! ✓
• Is \(\mathrm{x/2 = 2 \times (x/4)}\)? Yes! ✓

4. Final Answer

The number of men at the picnic is \(\mathrm{x/4}\).

This corresponds to answer choice C.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "3 times as many adults as children"
Students often confuse the direction of this relationship. They might think it means "\(\mathrm{adults = children ÷ 3}\)" instead of "\(\mathrm{adults = 3 \times children}\)". This fundamental misunderstanding of the multiplier relationship will lead to completely incorrect variable setup.

2. Forgetting that adults = men + women
Students may treat men, women, adults, and children as four separate, independent groups instead of recognizing that adults is composed of men and women. This leads to setting up equations with four variables instead of understanding the hierarchical relationship.

3. Starting with the wrong base variable
Since the question asks for men in terms of x, students might try to start by defining children or women as their base variable, making the algebra unnecessarily complex and increasing the chance of errors in the relationship setup.

Errors while executing the approach

1. Algebraic errors when expressing children in terms of men
When students have "\(\mathrm{adults = 3M}\)" and "\(\mathrm{adults = 3 \times children}\)", they may incorrectly solve for children, getting "\(\mathrm{children = 3M}\)" instead of "\(\mathrm{children = M}\)". This happens when they don't properly cancel the coefficient of 3 on both sides.

2. Arithmetic mistakes when adding fractions in verification
When checking their work with \(\mathrm{x/4 + x/2 + x/4 = x}\), students often struggle with adding fractions, particularly converting \(\mathrm{x/2}\) to \(\mathrm{2x/4}\), leading them to doubt their correct answer.

3. Setting up the total equation incorrectly
Students may write the total as "\(\mathrm{3M + M = x}\)" (thinking adults + children) instead of "\(\mathrm{M + 2M + M = x}\)" (men + women + children), forgetting to break down adults into its constituent parts.

Errors while selecting the answer

1. Solving for the wrong quantity
After finding \(\mathrm{M = x/4}\), students might accidentally select the answer choice corresponding to women (\(\mathrm{x/2}\)) or adults (\(\mathrm{3x/4}\)) instead of men, especially if they calculated these quantities during their verification process.

2. Inverting the final fraction
From the equation \(\mathrm{4M = x}\), students might incorrectly solve to get \(\mathrm{M = 4x}\) instead of \(\mathrm{M = x/4}\), then try to match 4x to one of the given fractional answer choices, leading to confusion.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a strategic smart number for the total

Since we need to work with relationships involving multiples of 2 and 3, let's choose \(\mathrm{x = 12}\) people total. This number works well because it's divisible by the factors we'll encounter.

Step 2: Set up the relationships with our smart number

Let M = number of men
Then: \(\mathrm{Women = 2M}\) (twice as many women as men)
\(\mathrm{Children = C}\)
\(\mathrm{Adults = M + 2M = 3M}\)
Since \(\mathrm{adults = 3 \times children: 3M = 3C}\), so \(\mathrm{C = M}\)

Step 3: Express total in terms of M

\(\mathrm{Total = Men + Women + Children}\)
\(\mathrm{12 = M + 2M + M = 4M}\)
Therefore: \(\mathrm{M = 12 ÷ 4 = 3}\) men

Step 4: Verify our relationships work

• Men = 3
• Women = \(\mathrm{2 \times 3 = 6}\)
• Children = 3
• Adults = \(\mathrm{3 + 6 = 9}\)
• Check: \(\mathrm{9 \text{ adults} = 3 \times 3 \text{ children}}\) ✓
• Check: \(\mathrm{6 \text{ women} = 2 \times 3 \text{ men}}\) ✓
• Total = \(\mathrm{3 + 6 + 3 = 12}\) ✓

Step 5: Find the pattern

With \(\mathrm{x = 12}\), we have 3 men.
The ratio is \(\mathrm{3/12 = 1/4}\)
Therefore: Number of men = \(\mathrm{x/4}\)

Answer: C. \(\mathrm{x/4}\)